AFAIK the christian and druse (religious sect with a mix of Islam, Bhuddism and Christianity living in all countries in the area. The druse men mostly are professional soldiers in the Lebanese, Israeli and Syrian armies) dominated, israeli and french trained lebanese army fights a sunni fundamentalist group in one or serveral "refugee camps"; actually now small cities of palestinians in Lebanon, as these camps are there since 1949.
These militant groups are NOT Hezbollah, as they are sunni/palestinian and not lebanese shiite. They may share the common goal of destroying Israel, but as always with Arabs, slaughtering each other comes before slaughtering your real enemy
Apparently the lebanese have to do the dirty work for the IDF now.